Sici Spiral

The curve (Si(t), Ci(t)) traced by the sine and cosine integrals

Goal of this page

Understand the parametric equations $(x,y) = (\mathrm{Si}(t),\, \mathrm{Ci}(t))$ of the Sici spiral, the definitions of the sine and cosine integrals, and the way the curve spirals in toward its limit point.

1. Definition and the integral functions

The Sici spiral is defined as the parametric curve whose coordinates are the sine integral (Si) and the cosine integral (Ci).

Sine integral and cosine integral

$$\begin{aligned} \mathrm{Si}(t) &= \int_0^t \frac{\sin u}{u}\,du \\ \mathrm{Ci}(t) &= -\int_t^\infty \frac{\cos u}{u}\,du = \gamma + \ln t + \int_0^t \frac{\cos u - 1}{u}\,du \end{aligned}$$

Here $\gamma \approx 0.5772$ is the Euler–Mascheroni constant.

Parametric equations of the Sici spiral

$$x(t) = \mathrm{Si}(t), \quad y(t) = \mathrm{Ci}(t) \quad (t > 0)$$
$\mathrm{Si}(t)$
$\mathrm{Ci}(t)$
$\left(\tfrac{\pi}{2},\,0\right)$
$t\to 0^+$
$\mathrm{Ci}\to-\infty$
Figure 1: Sketch of the Sici spiral $(\mathrm{Si}(t),\,\mathrm{Ci}(t))$. As $t \to 0^+$, $\mathrm{Si}\to 0$ and $\mathrm{Ci}\to-\infty$ (downward along the $y$-axis); as $t \to \infty$ the curve spirals in to the point $(\pi/2,\, 0)$.

2. Properties

Main properties

  • Limit point: as $t \to +\infty$, $(x,y) \to (\pi/2,\, 0)$.
  • $t \to 0^+$: $\mathrm{Si}(0) = 0$, while $\mathrm{Ci}(t) \to -\infty$ (the curve runs off to negative infinity along the $y$-axis).
  • Tangent direction: from $dx/dt = \sin t / t$ and $dy/dt = \cos t / t$, the slope of the tangent is $\tan(\text{angle}) = \cos t / \sin t = \cot t$.
  • Loops: $x$ has extrema where $\sin t = 0$ ($t = n\pi$) and $y$ has extrema where $\cos t = 0$; the curve oscillates as it converges to its limit point.

3. Numerical examples

Values at representative $t$

$t$ $\mathrm{Si}(t)$ $\mathrm{Ci}(t)$
$\pi/2 \approx 1.571$$1.371$$0.472$
$\pi \approx 3.142$$1.852$$0.074$
$2\pi \approx 6.283$$1.418$$-0.023$
$\infty$$\pi/2 \approx 1.571$$0$

$\mathrm{Si}(t)$ oscillates around $\pi/2$ as it converges, while $\mathrm{Ci}(t)$ converges to $0$.

FAQ

Q1. What is the Sici spiral?

The Sici spiral is the parametric curve $(x,y)=(\mathrm{Si}(t),\mathrm{Ci}(t))$ whose $x$- and $y$-coordinates are the sine integral $\mathrm{Si}(t)$ and the cosine integral $\mathrm{Ci}(t)$. As $t$ runs from $0$ to infinity, the curve spirals in toward the point $(\pi/2, 0)$.

Q2. What are the sine integral Si(t) and cosine integral Ci(t)?

The sine integral is defined by $\displaystyle \mathrm{Si}(t) = \int_0^t (\sin u)/u\,du$ and the cosine integral by $\displaystyle \mathrm{Ci}(t) = -\int_t^\infty (\cos u)/u\,du$. Both arise from integrals of the sinc function and appear in optics and signal processing.

Q3. How does the Sici spiral differ from the Cornu spiral?

The Cornu spiral is the curve parametrized by the Fresnel integrals $(C(t), S(t))$, whose curvature is proportional to arc length. The Sici spiral uses the sine and cosine integrals; it likewise converges in loops but follows a different law of curvature. Both appear in wave optics and diffraction calculations.

Related topics and references

References